How can I move an object with a UV map to a DIFFERENT project?
From what i understand you want to move the image texture connected to the object's material to your new project file. There's two ways i know how to do that.

You can go to your old project, open the UV/Image editor tab, select your image and click Image > Save As Image. You can then bring this image into your new project from the location where you saved it at.



Another option is to append the files from your old project into your new one. You can do this by pressing Shift+F1 or by going to File > Append. From there you must locate your old .blend and find the required files in the corresponding folders.



I hope this was useful to you. I'm sorry if i misread your question, feel free to correct me.